Hi Jonafly,
Yes, those do exist, they are called 'wildlife hospitals' in England, and 'rehab centers' (from 'rehabilitation') in the USA.
I think some do show people round, but it is a difficult thing for them to do - first, a lot of the birds they have are very timid, and to let people get close to see them, may cause them distress. They recover better if they don't have to worry about people looking at them (some places, they even take care that the staff remain invisible to the birds, by feeding them from behind covers, etc). Also, the staff tend to be very busy helping the birds (mostly, they are volunteers), and don't have time to show people round.
They may have occasional open days when there are extra volunteers to show people round. Some also have 'long term resident' birds, e.g. with a wing missing and so impossible to release; these birds may become more accustomed to people and can be shown.
Michael